The following passage is taken from NDSolve Overview in Mathemaitca 5.1:
> Here is a simple DAE.
>
> In[119]:=
> NDSolve[{Derivative[2][x][t] + y[t] == x[t],
> x[t]^2 + y[t]^2 == 1, x[0] == 0,
> Derivative[1][x][0] == 1}, {x, y}, {t, 0, 2}]
>
> From In[119]:=
> NDSolve::ndsz: At t == 1.6656481721762024`, step size is effectively
> zero; \
> singularity or stiff system suspected.
>
> Out[119]=
> {{x -> InterpolatingFunction[],
> y -> InterpolatingFunction[]}}
>
> Note that while both of the equations have derivative terms, the
> variable y appears without any derivatives, so NDSolve issues a
> warning message. When the usual substitution to convert to first
> order equations is made, one of the equations does indeed become
> effectively algebraic.
It is not true that both equations have derivative terms nor does the
warning message seem to be of the kind that is referred to. If we
change the equations and initial conditions to something like this:
NDSolve[{Derivative[2][x][t] + y[t] == x[t],
Derivative[1][x][t]^2 + y[t]^2 == 1, x[0] == 0,
Derivative[1][x][0] == 0}, {x, y}, {t, 0, 2}]
we get a message more like the one that is referred to in the
documentation.
NDSolve::pdord: Some of the functions have zero differential order so
the equations will be solved as a system of differential-algebraic
equations.
NDSolve::ndsz: At `1.0525780991716984 step size is effectively \
zero; singularity or stiff system suspected.
{{x -> InterpolatingFunction[],
y -> InterpolatingFunction[]}}
Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/andrzej/index.htmlhttp://www.mimuw.edu.pl/~akoz/